Solve H(3) For H(x) = 2x² + 2x - 2 A Step-by-Step Guide
In this article, we will delve into the process of evaluating a quadratic function at a specific point. Specifically, we will focus on the function h(x) = 2x² + 2x - 2 and determine the value of h(3). This involves substituting x with 3 in the function and simplifying the expression to arrive at the final answer. Understanding how to evaluate functions is a fundamental concept in algebra and calculus, and this step-by-step solution will provide a clear and concise guide to the process.
Understanding Function Evaluation
At its core, function evaluation is the process of determining the output value of a function for a given input value. In mathematical terms, a function establishes a relationship between a set of inputs and a set of permissible outputs. When we evaluate a function at a specific input, we are essentially finding the corresponding output value that the function associates with that particular input. This concept is crucial for analyzing and understanding the behavior of functions, as it allows us to predict how a function will respond to different inputs.
The Concept of Function Evaluation Explained
The concept of function evaluation is fundamental to understanding how mathematical functions work. A function, in essence, is a rule or a process that takes an input value, performs some operations on it, and produces an output value. Think of it like a machine: you feed it an input, and it spits out an output based on its internal workings. When we evaluate a function at a particular value, we are essentially asking, "What output will this function produce if we give it this specific input?"
For example, consider the function f(x) = x + 2. This function tells us to take any input x, add 2 to it, and the result is the output. So, if we want to evaluate f(2), we substitute x with 2: f(2) = 2 + 2 = 4. This means that when the input is 2, the output of the function is 4. Function evaluation is a cornerstone of mathematics, used extensively in algebra, calculus, and various other branches. It allows us to analyze, predict, and manipulate the behavior of functions, which are mathematical representations of real-world phenomena.
Why Function Evaluation Matters
Function evaluation is not just a theoretical exercise; it has immense practical significance. It's the tool we use to make predictions, solve problems, and understand relationships in various fields. Imagine you have a function that models the trajectory of a ball thrown in the air. By evaluating this function at different time values, you can determine the ball's height at those times. Or, consider a function that represents the growth of a population. Evaluating it at different years can give you insights into the population's size over time. In engineering, function evaluation is crucial for designing structures, circuits, and systems. By evaluating functions that model physical properties, engineers can optimize designs and ensure safety.
Moreover, function evaluation is essential in data analysis and machine learning. We often use functions to model patterns in data, and evaluating these functions helps us make predictions and draw conclusions. For instance, a function might model customer behavior, and evaluating it can help a business forecast demand. Function evaluation is also the foundation of computer programming. When you write code, you are essentially defining functions that the computer will evaluate. Each time the program runs, these functions are evaluated with specific inputs, leading to the desired output. In short, function evaluation is a fundamental skill that empowers us to understand and interact with the world around us, making it an indispensable tool in mathematics and beyond.
The Quadratic Function h(x) = 2x² + 2x - 2
The function we are working with, h(x) = 2x² + 2x - 2, is a quadratic function. Quadratic functions are a specific type of polynomial function characterized by the highest power of the variable being 2. They have a distinctive U-shaped curve when graphed, known as a parabola. The coefficients in a quadratic function play a crucial role in determining the shape and position of the parabola. In our function, the coefficient of the x² term is 2, the coefficient of the x term is also 2, and the constant term is -2. Understanding the properties of quadratic functions is essential in various fields, including physics, engineering, and economics, as they often model real-world phenomena such as projectile motion, optimization problems, and growth patterns.
Dissecting the Components of the Function
To truly grasp how to evaluate h(x) = 2x² + 2x - 2, let's break it down into its individual components. The function is composed of three terms: 2x², 2x, and -2. Each term contributes differently to the overall behavior of the function. The term 2x² is the quadratic term, and it's the driving force behind the parabolic shape of the function's graph. The coefficient 2 in front of x² scales the parabola, making it narrower or wider. A larger coefficient makes the parabola narrower, while a smaller coefficient makes it wider. The term 2x is the linear term, and it introduces a slant to the parabola. It influences the position of the parabola's vertex, which is the point where the parabola changes direction. The coefficient 2 here also affects the steepness of this slant. Finally, the constant term -2 shifts the entire parabola vertically. It determines the y-intercept of the graph, which is the point where the parabola crosses the y-axis. Understanding these components is crucial not just for evaluation but also for analyzing the function's behavior, finding its roots, and graphing it. By dissecting the function, we gain a deeper insight into its nature and how it transforms inputs into outputs.
The Significance of Quadratic Functions
Quadratic functions hold a place of significance in mathematics and its applications due to their ability to model a wide array of real-world phenomena. Their U-shaped parabolic curves accurately represent scenarios involving projectile motion, such as the path of a ball thrown in the air or the trajectory of a rocket. The highest or lowest point on the parabola, known as the vertex, often corresponds to a maximum or minimum value in these situations, like the maximum height of a projectile or the minimum cost in an optimization problem. In engineering, quadratic functions are used to design arches, bridges, and other structures, as their parabolic shape provides optimal strength and stability. The curves formed by suspension cables on bridges are often modeled using quadratic functions. In economics, quadratic functions can represent cost curves, revenue curves, and profit curves. The vertex of these parabolas can help businesses determine the optimal production level to maximize profit or minimize costs.
Furthermore, quadratic functions play a crucial role in optimization problems. Many real-world scenarios involve finding the best possible solution, such as maximizing profit, minimizing cost, or optimizing resource allocation. Quadratic functions provide a mathematical framework for modeling these scenarios, and techniques like finding the vertex or roots of the quadratic equation allow us to identify the optimal solutions. From the trajectory of a baseball to the design of a satellite dish, quadratic functions are essential tools for understanding and solving problems in a vast range of fields.
Solving for h(3): The Step-by-Step Process
Now, let's get to the heart of the matter: solving for h(3). This means substituting x with 3 in the function h(x) = 2x² + 2x - 2 and then simplifying the resulting expression. We will follow the order of operations (PEMDAS/BODMAS) to ensure we arrive at the correct answer. This process involves first squaring the value of 3, then multiplying by the coefficient 2, followed by multiplying 2 by 3, and finally adding and subtracting the terms to obtain the final result. This methodical approach is crucial for accurate function evaluation.
Step 1: Substitution
The first and most crucial step in evaluating h(3) is the substitution. This involves replacing every instance of the variable x in the function with the value 3. So, we take our function h(x) = 2x² + 2x - 2 and substitute x with 3. This transforms the function into an expression involving only numerical values, which we can then simplify. The substitution step sets the stage for the rest of the calculation, so it's vital to perform it accurately. Errors in substitution can propagate through the entire solution, leading to an incorrect final answer. It's like setting up the foundation for a building; if the foundation is flawed, the rest of the structure will be unstable.
In our case, after substituting x with 3, we get h(3) = 2(3)² + 2(3) - 2. Notice how every x has been replaced by a 3, and the parentheses are crucial to maintain the correct order of operations. The next steps involve simplifying this numerical expression using the rules of arithmetic, but the foundation has been laid with the correct substitution. The substitution step is not just about replacing a variable with a number; it's about transforming the function into a specific instance, a snapshot of the function's output when the input is 3. This step bridges the gap between the abstract function and a concrete numerical value, allowing us to make predictions, solve problems, and understand the function's behavior at a particular point.
Step 2: Applying the Order of Operations
Once we have successfully substituted x with 3, we are left with a numerical expression that needs to be simplified. This is where the order of operations comes into play. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), is a set of rules that dictate the sequence in which mathematical operations should be performed. Following this order ensures that we arrive at the correct answer, regardless of the complexity of the expression. Failing to adhere to the order of operations can lead to ambiguity and incorrect results. It's like following a recipe; if you mix the ingredients in the wrong order, the final dish won't turn out as expected. In mathematics, the order of operations is the recipe for simplifying expressions, and it's essential for consistent and accurate calculations.
In our expression, h(3) = 2(3)² + 2(3) - 2, we first need to deal with the exponent. According to PEMDAS/BODMAS, exponents come before multiplication, addition, and subtraction. So, we calculate 3² first, which is 3 multiplied by itself, resulting in 9. Now our expression becomes h(3) = 2(9) + 2(3) - 2. The next operations are multiplications. We have two multiplications to perform: 2 multiplied by 9 and 2 multiplied by 3. Performing these multiplications, we get 18 and 6, respectively. Our expression now simplifies to h(3) = 18 + 6 - 2. Finally, we have addition and subtraction, which are performed from left to right. We first add 18 and 6, which gives us 24. Then, we subtract 2 from 24, resulting in 22. Therefore, after applying the order of operations step-by-step, we have simplified the expression and arrived at the final answer: h(3) = 22. The order of operations is not just a set of rules; it's a systematic approach to simplifying expressions, ensuring clarity and consistency in mathematical calculations. By following PEMDAS/BODMAS, we can break down complex expressions into manageable steps, arriving at the correct solution every time.
Step 3: Simplification and the Final Result
After applying the order of operations, we arrive at the simplified expression that gives us the value of h(3). In our case, following the steps of substitution and the order of operations, we have transformed the original function evaluation problem into a straightforward calculation. This final step solidifies our understanding of the function's behavior at the specific point x = 3. The result, h(3) = 22, tells us that when the input to the quadratic function h(x) = 2x² + 2x - 2 is 3, the output is 22. This single numerical value encapsulates the function's response to that particular input, providing a concrete data point that can be used for analysis, graphing, or further calculations.
The simplification process is not just about arriving at a numerical answer; it's about unraveling the function's behavior. Each step in the simplification process reveals a layer of the function's transformation, from the initial substitution to the final result. By carefully following the order of operations, we ensure that we are accurately representing the function's inherent logic. The final result, h(3) = 22, is the culmination of this process, a concise representation of the function's output for the given input. This value can be interpreted in various ways, depending on the context of the problem. For instance, if h(x) represents the height of a projectile at time x, then h(3) = 22 would mean that the projectile is at a height of 22 units at time 3. The simplification and final result are not just the end of the calculation; they are the gateway to understanding the function's behavior and its implications in real-world scenarios. By arriving at h(3) = 22, we have not only solved the problem but also gained a valuable insight into the quadratic function h(x).
The Final Solution: h(3) = 22
Therefore, after meticulously substituting x with 3 in the function h(x) = 2x² + 2x - 2 and diligently applying the order of operations, we arrive at the final solution: h(3) = 22. This result represents the value of the function h(x) when x is equal to 3. It signifies a specific point on the graph of the quadratic function, and it provides a concrete answer to the problem posed. This step-by-step solution demonstrates the process of function evaluation, a fundamental concept in mathematics that is essential for understanding the behavior of functions and their applications in various fields. The final solution, h(3) = 22, is not just a number; it's the culmination of a logical and methodical process, a testament to the power of mathematical reasoning.
Detailed Breakdown of the Solution
Let's recap the entire solution process in a detailed breakdown to ensure clarity and understanding. We started with the quadratic function h(x) = 2x² + 2x - 2 and the task of evaluating h(3). The first step was substitution, where we replaced every instance of x in the function with the value 3. This transformed the function into h(3) = 2(3)² + 2(3) - 2. This substitution step is crucial as it sets the stage for the numerical calculation.
Next, we applied the order of operations (PEMDAS/BODMAS) to simplify the expression. First, we dealt with the exponent: 3² equals 9, so the expression became h(3) = 2(9) + 2(3) - 2. Then, we performed the multiplications: 2 multiplied by 9 is 18, and 2 multiplied by 3 is 6. This simplified the expression to h(3) = 18 + 6 - 2. Finally, we carried out the addition and subtraction from left to right: 18 plus 6 is 24, and 24 minus 2 is 22. Thus, we arrived at the final solution: h(3) = 22. This step-by-step breakdown highlights the importance of each step in the process, from the initial substitution to the final arithmetic operations. It underscores the need for precision and adherence to mathematical rules to arrive at the correct solution.
Implications of the Solution
The solution h(3) = 22 carries significant implications when we consider the quadratic function h(x) = 2x² + 2x - 2 in its broader context. This result represents a specific point on the parabola that is the graph of the function. The point (3, 22) lies on this parabola, and it provides valuable information about the function's behavior. For instance, we know that at x = 3, the function's output is 22, which can be interpreted as the y-coordinate on the graph. This point can be used to sketch the parabola, analyze its symmetry, and understand its overall shape.
Furthermore, the solution can have practical interpretations depending on the context of the problem. If h(x) represents a physical quantity, such as the height of a projectile at time x, then h(3) = 22 would mean that at time 3, the projectile is at a height of 22 units. If h(x) represents a cost function, then h(3) = 22 would indicate the cost incurred at a certain production level or activity level represented by x = 3. The solution h(3) = 22 is not just a numerical answer; it's a data point that can be used to model, predict, and understand the real-world phenomena represented by the function. It's a bridge between the abstract mathematical concept and its concrete applications, highlighting the power of function evaluation in problem-solving and decision-making. By understanding the implications of the solution, we can gain deeper insights into the quadratic function and its role in various scenarios.
Conclusion: Mastering Function Evaluation
In conclusion, evaluating h(3) for the quadratic function h(x) = 2x² + 2x - 2 has been a journey through the fundamental principles of function evaluation. We have meticulously navigated the steps of substitution, the order of operations, and simplification to arrive at the final solution: h(3) = 22. This process has not only provided us with a numerical answer but also deepened our understanding of how functions work and how they can be applied in various contexts. Mastering function evaluation is a crucial skill in mathematics, as it forms the basis for more advanced concepts and problem-solving techniques. The ability to evaluate functions accurately and efficiently empowers us to analyze, predict, and manipulate mathematical models, making it an indispensable tool in various fields, from science and engineering to economics and computer science.
Key Takeaways from the Solution
As we conclude this detailed exploration of evaluating h(3) for the quadratic function h(x) = 2x² + 2x - 2, it's crucial to highlight the key takeaways from the solution process. These takeaways serve as valuable reminders and guidelines for tackling similar problems in the future, reinforcing our understanding of function evaluation and its nuances. Firstly, the importance of accurate substitution cannot be overstated. Replacing every instance of the variable x with the given value is the foundation of the entire process, and any error at this stage will inevitably lead to an incorrect final answer. The substitution must be performed meticulously, ensuring that all occurrences of x are correctly replaced.
Secondly, the order of operations (PEMDAS/BODMAS) is the guiding principle for simplifying expressions. It dictates the sequence in which mathematical operations should be performed, ensuring consistency and accuracy in calculations. Adhering to this order is not just a matter of convention; it's a fundamental requirement for arriving at the correct solution. Thirdly, each step in the simplification process should be performed with precision and care. From handling exponents to carrying out multiplications, additions, and subtractions, every operation must be executed accurately to maintain the integrity of the solution. Errors in any step can compound and lead to a wrong final answer.
Finally, the solution h(3) = 22 is not just a numerical value; it represents a specific point on the graph of the function, and it can have practical interpretations depending on the context of the problem. Understanding the implications of the solution is as important as arriving at the correct answer. By keeping these key takeaways in mind, we can approach function evaluation problems with confidence and competence, ensuring that we not only arrive at the correct solutions but also gain a deeper understanding of the underlying mathematical concepts. The takeaways are not just about mechanics; they are about building a robust problem-solving approach, which is a crucial skill in mathematics and beyond.
Further Exploration of Quadratic Functions
Having successfully evaluated h(3) for the quadratic function h(x) = 2x² + 2x - 2, it's natural to consider further exploration of quadratic functions and their properties. Function evaluation is just one aspect of understanding quadratic functions; there's a wealth of knowledge to be gained by delving into other facets, such as graphing, finding roots, determining the vertex, and analyzing the discriminant. Graphing a quadratic function provides a visual representation of its behavior, allowing us to observe its parabolic shape, symmetry, and key features like the vertex and intercepts. The vertex, which is the highest or lowest point on the parabola, is particularly significant as it represents the maximum or minimum value of the function.
Finding the roots of a quadratic function involves solving the equation h(x) = 0. The roots are the x-intercepts of the graph, and they provide valuable information about the function's behavior and its relationship to the x-axis. The discriminant, which is the part of the quadratic formula under the square root, tells us about the nature of the roots: whether they are real or complex, distinct or repeated. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one repeated real root, and a negative discriminant indicates two complex roots.
Furthermore, understanding the coefficients of the quadratic function is crucial. The coefficient of the x² term determines the parabola's concavity (whether it opens upwards or downwards) and its width. The coefficient of the x term influences the position of the vertex, and the constant term determines the y-intercept. By exploring these aspects of quadratic functions, we gain a comprehensive understanding of their behavior and their applications in various fields. From projectile motion in physics to optimization problems in economics, quadratic functions are powerful tools for modeling and solving real-world problems. Further exploration of quadratic functions opens up a world of mathematical insights and practical applications, making it a worthwhile endeavor for anyone interested in deepening their understanding of mathematics.
